We present a symbolic algorithm generating finite-element schemes with interpolating Hermite polynomials intended for solving the boundary-value problems with self-adjoint second-order differential equation and implemented in the Maple computer algebra system. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicity are derived. The integrals of interpolating Hermite polynomials are used for constructing the stiffness and mass matrices and formulating a generalized algebraic eigenvalue problem. The algorithm is used to generate Fortran routines that allow solution of the generalized algebraic eigenvalue problem with matrices of large dimension. The efficiency of the programs generated in Maple and Fortran is demonstrated by the examples of exactly solvable quantum-mechanical problems with continuous and piecewise continuous potentials.